Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Solvability of a fractional Cauchy problem based on modified fixed point results of non-compactness measures

1 Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore-632014, TN, India
2 Department of Mathematics and Applied Mathematics, University of Johannesburg, Kingsway Campus, Auckland Park 2006, South Africa
3 Cloud Computing Center, University Malaya, Malaysia

Special Issues: Initial and Boundary Value Problems for Differential Equations

We study the solvability of a fractional Cauchy problem based on new development of fixed point theorem, where the operator is suggested to be non-compact on its domain. Moreover, we shall prove that the solution is bounded by a fractional entropy (entropy solution). For this purpose, we establish a collection of basic fixed point results, which generalizes and modifies some well known results. Our attention is toward the concept of a measure of non-compactness to generalize µ-set contractive condition, using three control functions.
  Article Metrics

Keywords fractional calculus; fractional differential operator; fractional differential equation; fixed point theorem; measure of non-compactness; entropy solution

Citation: Hemant Kumar Nashine, Rabha W. Ibrahim. Solvability of a fractional Cauchy problem based on modified fixed point results of non-compactness measures. AIMS Mathematics, 2019, 4(3): 847-859. doi: 10.3934/math.2019.3.847


  • 1.R. P. Agarwal, M. Benchohra and D. Seba, On the application of measure of noncompactness to the existence of solutions for fractional differential equations, Results Math., 55 (2009), 221-230.    
  • 2.R. P. Agarwal, M. Meehan and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, 2001.
  • 3.A. Aghajani, J. Banas and N. Sabzali, Some generalizations of Darbo fixed point theorem and applications, Bull. Belg. Math. Soc. Simon Stevin., 20 (2013), 345-358.
  • 4.A. Aghajani, J. Banas and Y. Jalilian, Existence of solution for a class of nonlinear Volterra singular integral equations, Comp. Math. Appl., 62 (2011), 1215-1227.    
  • 5.A. Aghajani and N. Sabzali, A coupled fixed point theorem for condensing operators with application to system of integral equations, J. Nonlinear Convex Anal., 15 (2014), 941-952.
  • 6.A. Aghajani, R. Allahyari and M. Mursaleen, A generalization of Darbo's theorem with application to the solvability of systems of integral equations, J. Comput. Appl. Math., 260 (2014), 68-77.    
  • 7.R. Arab, Some fixed point theorems in generalized Darbo fixed point theorem and the existence of solutions for system of integral equations, J. Korean Math. Soc., 52 (2015), 125-139.    
  • 8.R. Arab, The existence of fixed points via the measure of noncompactness and its application to functional-integral equations, Mediterr. J. Math., 13 (2016), 759-773.    
  • 9.J. Banas, Measures of noncompactness in the space of continuous tempered functions, Demonstr. Math., 14 (1981), 127-133.
  • 10.J. Banas and K. Goebel, Measures of noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, New York: Dekker, 1980.
  • 11.L. Byszewski, Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem, J. Math. Anal. Appl., 162 (1991), 494-505.    
  • 12. G. Darbo, Punti uniti in transformazion a condominio non compatto, Rend. Sem. Math. Univ. Padova, 24 (1955), 84-92.
  • 13.M. M. El-Borai, Semigroups and some nonlinear fractional differential equations, Appl. Math. Comput., 149 (2004), 823-831.
  • 14.M. A. A. El-Sayeed, Fractional order diffusion wave equation, Int. J. Theor. Phys., 35 (1996), 311-322.    
  • 15.D. Guo, V. Lakshmikantham and X. Liu, Nonlinear Integral Equations in Abstract Spaces, vol. 373 of Mathematics and Its Applications, Dordrecht, The Netherlands : Kluwer Academic Publishers, 1996.
  • 16.R. W. Ibrahim and H. A. Jalab, Existence of entropy solutions for nonsymmetric fractional systems, Entropy, 16 (2014), 4911-4922.    
  • 17.R. W. Ibrahim and H. A. Jalab, Existence of Ulam stability for iterative fractional differential equations based on fractional entropy, Entropy, 17 (2015), 3172-3181.    
  • 18.O. K. Jaradat, A. Al-Omari and S. Momani, Existence of the mild solution for fractional semilinear initial value problem, Nonlinear Anal., 69 (2008), 3153-3159.    
  • 19.A. Kilbas, and S. Marzan, Cauchy problem for differential equation with Caputo derivative, Fract. Calc. Appl. Anal., 7 (2004), 297-321.
  • 20.M. Mursaleen and S. A. Mohiuddine, Applications of measures of noncompactness to the infinite system of differential equations in lp spaces, Nonlinear Anal. Theory, Methods Appl., 75 (2012), 2111-2115.    
  • 21.M. Mursaleen and A. Alotaibi, Infinite system of differential equations in some BK spaces, Abstr. Appl. Anal., 2012 (2012), Article ID 863483.
  • 22.C. Tsallis, Generalized entropy-based criterion for consistent testing, Phys. Rev. E, 58 (1998), 1442.


Reader Comments

your name: *   your email: *  

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved